# Orthomodular lattices can be converted into left residuated l-groupoids

@article{Chajda2017OrthomodularLC, title={Orthomodular lattices can be converted into left residuated l-groupoids}, author={Ivan Chajda and Helmut Langer}, journal={Miskolc Mathematical Notes}, year={2017}, volume={18}, pages={685} }

We show that every orthomodular lattice can be considered as a left residuated l-groupoid satisfying divisibility, antitony, the double negation law and three more additional conditions expressed in the language of residuated structures. Also conversely, every left residuated l-groupoid satisfying the mentioned conditions can be organized into an orthomodular lattice.

## 15 Citations

Negative Translations of Orthomodular Lattices and Their Logic

- Mathematics, PhysicsElectronic Proceedings in Theoretical Computer Science
- 2021

We introduce residuated ortholattices as a generalization of—and environment for the investigation of—orthomodular lattices. We establish a number of basic algebraic facts regarding these structures,…

Left residuated lattices induced by lattices with a unary operation

- Computer Science, MathematicsSoft Comput.
- 2020

A variety of lattices with a unary operation which contains exactly those lattices which can be converted into a left residuated lattice by use of the above mentioned operations.

On residuation in paraorthomodular lattices

- Computer Science, MathematicsSoft Comput.
- 2020

This paper starts the investigation of material implications in paraorthomodular lattices by showing that any bounded modular lattice with antitone involution A can be converted into a left-residuated groupoid if it satisfies a strengthened form of regularity.

Residuated Structures and Orthomodular Lattices

- Computer Science, MathematicsStud Logica
- 2021

This work investigates the lattice of subvarieties of pointed left-residuated, develops a theory of left nuclei, and extends some parts of the theory of join-completions of residuated to the left- Residuated case, giving a new proof of MacLaren’s theorem for orthomodular lattices.

Residuation in modular lattices and posets

- MathematicsAsian-European Journal of Mathematics
- 2019

We show that every complemented modular lattice can be converted into a left residuated lattice where the binary operations of multiplication and residuum are term operations. The concept of an…

How to introduce the connective implication in orthomodular posets

- Mathematics
- 2019

Since orthomodular posets serve as an algebraic axiomatization of the logic of quantum mechanics, it is a natural question how the connective of implication can be defined in this logic. It should be…

Residuated operators in complemented posets

- MathematicsAsian-European Journal of Mathematics
- 2018

Using the operators of taking upper and lower cones in a poset with a unary operation, we define operators [Formula: see text] and [Formula: see text] in the sense of multiplication and residuation,…

Left residuated operators induced by posets with a unary operation

- Mathematics, Computer ScienceSoft Comput.
- 2019

Modifications of so-called quantum structures, in particular orthomodular posets, like pseudo-orthomodULAR, pseudo-Boolean and Boolean posets are investigated here in order to show that they are operator left residuated or even operator residuated.

Residuation in finite posets

- Mathematics
- 2019

Abstract When an algebraic logic based on a poset instead of a lattice is investigated then there is a natural problem how to introduce implication to be everywhere defined and satisfying (left)…

Weakly Orthomodular and Dually Weakly Orthomodular Lattices

- Mathematics, Computer ScienceOrder
- 2018

It turns out that lattices being both weakly orthomodular and dually weakly Orthmodular are in fact complemented but the complementation need not be neither antitone nor an involution.

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